When Euler-Poisson-Darboux meets Painlevé and Bratu: On the numerical solution of nonlinear wave equations

Roland Glowinski (Department of Mathematics, University of Houston, Texas, U.S.A.; Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France)

Annalisa Quaini (Department of Mathematics, University of Houston, Texas, U.S.A.)

Abstract

The main goal of this article is to extend to Euler-Poisson-Darboux nonlinear wave equations the computational methods we employed in a previous work to solve a nonlinear equation coupling the classical wave operator with the nonlinear forcing term of the Painlevé I ordinary differential equation. In order to handle the extra (dissipative) term with singular coefficient encountered in the Euler-Poisson-Darboux equations, we advocate a five stage symmetrized operator-splitting scheme for the time-discretization. This scheme, combined with a finite element space discretization and adaptive time-stepping to monitor possible blow-up of the solution, provides a robust and accurate solution methodology, as shown by the results of the numerical experiments reported here. The nonlinearities we have considered are those encountered in the Painlevé I and II equations (and close variants of them), and the exponential one encountered in the celebrated Bratu problem.

Keywords

Euler-Poisson-Darboux nonlinear wave equations, Painlevé equations, Bratu problem, blow-up solutions, operator-splitting

2010 Mathematics Subject Classification

35L70, 65N30

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Published 16 April 2014